In a blender, combine half of the cucumber slices and the fresh mint leaves with 1 cup of cold water. Blend until smooth.
Strain the cucumber and mint mixture through a fine mesh sieve into a pitcher, pressing to extract as much liquid as possible.
Add the freshly squeezed lemon juice and sugar or honey to the pitcher. Stir well until the sweetener dissolves completely.
Pour in the remaining 3 cups of cold water and mix thoroughly. Taste and adjust sweetness or tartness as desired.
Add ice cubes to the pitcher or individual glasses to chill the lemonade.
Garnish with remaining cucumber slices, lemon slices, and mint sprigs before serving.
